We will investigate the interplay between the monopole Floer homology groups of \((M_{\varphi },\mathfrak {s}_L)\)  and the geometry of the theta characteristic \(L\) , specifically the space of its holomorphic sections \(H^0(L)\) . The latter is very well-studied object closely related to the study of bitangents in projective geometry [1]}; in general, its dimension \(h^0(L)\)  depends on the underlying holomorphic geometry of \(\Sigma \)  in a subtle way.
In our setup, because \(L\)  is \(\varphi \) -invariant, the action of \(\varphi \)  on \(\Sigma \)  lifts to an action \(\tilde{\varphi }\)  on \(L\)  (whose order is either \(d\)  or \(2d\) ) compatible with the identification \(L^2\cong K\) , and we obtain an induced finite order action \(\tilde{\varphi }^*\)  on \(H^0(L)\)  by pull-back. Our main result is the following.
